To log into Facebook without a code generator, you can either opt to receive a text/call, or approve a request from another device. Learn how!

Facebook notes that people who sign up for two-factor authentication using the U2F security key or code generator apps "do not need to register a phone number with Facebook." ...

More secure two-factor authentication methods, like code-generator apps and hardware tokens, have become popular ways to address this problem.

You can also use a code generator app and a U2F key for two-factor authentication and avoid giving your phone number to Facebook altogether.